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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '972.577', 'ambient_counter': 577, 'ambient_order': 972, 'ambient_tex': 'C_6^2.C_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 324, 'characteristic': True, 'core_order': 12, 'counter': 35, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '972.577.81.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '81.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '81.12', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 12, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 81, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3\\times \\He_3', 'simple': False, 'solvable': True, 'special_labels': ['S', 'L2', 'C4'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.577', 'aut_centralizer_order': 26244, 'aut_label': '81.a1', 'aut_quo_index': 24, 'aut_stab_index': 1, 'aut_weyl_group': '12.4', 'aut_weyl_index': 26244, 'centralizer': '3.b1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['27.a1', '27.b1', '27.c1', '27.d1'], 'contains': ['162.a1', '243.a1'], 'core': '81.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3611, 5085, 4354, 4053], 'generators': [27, 486, 324], 'label': '972.577.81.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '81.a1', 'normal_contained_in': ['27.a1', '27.b1'], 'normal_contains': ['243.a1', '324.a1'], 'normalizer': '1.a1', 'old_label': '81.a1', 'projective_image': '324.135', 'quotient_action_image': '3.1', 'quotient_action_kernel': '27.5', 'quotient_action_kernel_order': 27, 'quotient_fusion': None, 'short_label': '81.a1', 'subgroup_fusion': None, 'weyl_group': '3.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 1, 2], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_6', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [12, 18, 6], 'aut_gens': [[1, 3, 9, 54], [449, 345, 207, 705], [947, 345, 495, 306], [440, 669, 369, 735]], 'aut_group': None, 'aut_hash': 2759998136046393333, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 314928, 'aut_permdeg': 648, 'aut_perms': 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'aut_phi_ratio': 972.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 3, 3, 2], [3, 36, 18, 1], [6, 3, 2, 1], [6, 3, 6, 1], [6, 9, 3, 2], [9, 9, 6, 1], [18, 9, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.C_6^2.C_3^3.D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '27.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 27, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 3007605936380637636, 'autcentquo_nilpotent': False, 'autcentquo_order': 11664, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^3.D_6', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 1, 2], [3, 3, 8], [3, 36, 18], [6, 3, 8], [6, 9, 6], [9, 9, 6], [18, 9, 18]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '324.135', 'commutator_count': 1, 'commutator_label': '36.14', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 577, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 3, 2, 4], [3, 36, 2, 9], [6, 3, 2, 4], [6, 9, 2, 3], [9, 9, 2, 3], [18, 9, 2, 9]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1560, 'exponent': 18, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[9, 0, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '108.41', 'hash': 577, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 6, 6], 'inner_gens': [[1, 3, 846, 387], [1, 3, 9, 702], [190, 3, 9, 54], [694, 327, 9, 54]], 'inner_hash': 135, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': False, 'inner_tex': 'C_3^3:A_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 27], [3, 33], [9, 8]], 'label': '972.577', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6^2.C3^3', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 68, 'number_divisions': 35, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 54, 'number_subgroup_classes': 198, 'number_subgroups': 1440, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [3, 674], [6, 78], [9, 54], [18, 162]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 18, 'outer_gen_orders': [3, 6, 3, 3, 3, 2, 3], 'outer_gen_pows': [0, 0, 0, 0, 0, 360, 0], 'outer_gens': [[7, 327, 657, 738], [2, 327, 369, 387], [331, 651, 333, 381], [4, 3, 333, 72], [4, 3, 657, 54], [650, 651, 657, 645], [775, 39, 333, 90]], 'outer_group': '972.480', 'outer_hash': 480, 'outer_nilpotent': False, 'outer_order': 972, 'outer_permdeg': 12, 'outer_perms': [43545844, 43724429, 243, 80883, 80884, 83467469, 171866403], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^3:S_3^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13], [3, 1], [6, 16], [18, 4]], 'representations': {'PC': {'code': 13716770249610381885962533780839019, 'gens': [1, 2, 3, 5], 'pres': [7, 3, 3, 2, 3, 2, 3, 3, 17768, 58, 18651, 13549, 8201, 102, 32513, 6060, 166]}, 'Perm': {'d': 31, 'gens': [311078093176884317802042932110084, 584569054102336632813621557197440, 859593328609769681314613290542000, 1131865910908211202618153477254400, 1355364807847033521234373091499000, 7, 16]}}, 'schur_multiplier': [3, 3, 3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.C_3^3', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [9, 8, 4], 'aut_gens': [[2920, 1045, 1231, 784], [2974, 5371, 736, 784], [2920, 5392, 3178, 757], [5110, 5134, 5626, 784]], 'aut_group': '23328.dt', 'aut_hash': 3628243401347822157, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 27, 'aut_perms': [10030436526526919275772371736, 174626025659862093766770675, 585362951684203343265783677], 'aut_phi_ratio': 432.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 3, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:(S_3\\times \\GL(2,3))', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '486.183', 'autcent_hash': 183, 'autcent_nilpotent': False, 'autcent_order': 486, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^4:S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '48.29', 'autcentquo_hash': 29, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,3)', 'cc_stats': [[1, 1, 1], [3, 1, 8], [3, 3, 24]], 'center_label': '9.2', 'center_order': 9, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['27.3', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [3, 3, 2, 12]], 'element_repr_type': 'GLZq', 'elementary': 3, 'eulerian_function': 13, 'exponent': 3, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '27.5', 'hash': 12, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [1, 3, 3, 1], 'inner_gens': [[2920, 1045, 1231, 784], [2920, 1045, 1258, 784], [2920, 1018, 1231, 784], [2920, 1045, 1231, 784]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': False, 'inner_tex': 'C_3^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 27], [3, 6]], 'label': '81.12', 'linC_count': 108, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 27, 'linQ_dim': 8, 'linQ_dim_count': 27, 'linR_count': 27, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*He3', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 33, 'number_divisions': 17, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 56, 'number_subgroups': 104, 'old_label': None, 'order': 81, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [6, 6, 8], 'outer_gen_pows': [730, 730, 730], 'outer_gens': [[2920, 3181, 5365, 757], [5164, 733, 3238, 784], [2920, 736, 3208, 757]], 'outer_group': '2592.fv', 'outer_hash': 9019849488891309561, 'outer_nilpotent': False, 'outer_order': 2592, 'outer_permdeg': 12, 'outer_perms': [49367521, 182993909, 186219505], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times C_3^2:\\GL(2,3)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13], [6, 3]], 'representations': {'PC': {'code': 1088, 'gens': [1, 2, 3, 4], 'pres': [4, -3, 3, 3, -3, 258]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [1305911, 11495158, 36731395, 11225704]}, 'GLZq': {'d': 2, 'q': 9, 'gens': [757, 733, 1021, 2920]}, 'Perm': {'d': 12, 'gens': [51609744, 79974843, 135890884, 135890880]}}, 'schur_multiplier': [3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times \\He_3', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
